NFL Network Insider Ian Rapoport announced the visit for Gillislee after the latter was released by the Patriots on Saturday.
Gillislee would be looking to provide depth to the Saints' running backs corps with Mark Ingram set to serve a four-game suspension for violating the NFL's policy on performance-enhancing substances. Alvin Kamara, who was the NFL Offensive Rookie of the Year in 2017, is expected to lead the New Orleans backfield.
The 27-year-old Gillislee, who is listed at 5-foot-11 and 210 pounds, signed a two-year, $6.4 million contract with the Patriots as a restricted free agent in 2017. He had 104 carries for 383 yards and five touchdowns last season.
The Saints released Jonathan Williams and placed fellow running back Shane Vereen on injured reserve on Saturday. Rookie sixth-round draft pick Boston Scott is in line to serve as Kamara's backup for the first four weeks of the season.
Scott, who is listed at 5-6, had 25 carries for 121 yards and a touchdown in the preseason plus three catches for 18 yards.
New Orleans also has second-year running back Trey Edmunds and fullback Zach Line on the roster.